Overview

Raman processes

Introduction

In the preceding chapters, we mentioned several types of Raman processes. Their common feature is a resonant change of the energy of the photons that interact with the material system. The energy of the scattered photons may be lower (Stokes process) or higher (anti-Stokes) than that of the incident photons. The energy difference is transferred to the material system, where it must match an energy level separation. The photon energy itself, however, does not have to match exactly a transition frequency of the medium. This is commonly expressed by the statement that the Raman scattering proceeds through a virtual state, represented by the dashed line in Figure 7.1. The presence of a real state of the atom, indicated by the full line, nevertheless increases the coupling efficiency, as discussed in Chapter 3.

The earlier sections on three-level effects and optical anisotropy dealt with the mathematical formalism of Raman processes, using generic level systems to describe them.

Experimental arrangement

General considerations

Laser-induced dynamics

This chapter surveys the coherent evolution of coherences between angular momentum sublevels. Optical pumping excites this microscopic order, and it evolves under the influence of external magnetic fields and the laser radiation. The primary goal of this section is to show how the mathematical models developed in the preceding sections apply to real physical systems. We discuss how the observed signals arise and by which parameters the experimenter can control the dynamics of these systems.

Principle and overview

Phenomenology

Optical pumping (Happer 1972) is one of the earliest examples wherein optical radiation qualitatively modifies the properties of a material system. In its original implementation, it corresponds to a selective population of specific angular momentum states, starting from thermal equilibrium.

In the idealised process depicted in Figure 5.1, the light brings the atomic system from the initially disordered state, in which the populations of degenerate levels are equal, into an ordered state where the internal state of all atoms is the same. If we consider only the material system, it appears as if the evolution from the initially disordered state into an ordered state, where the population of one level is higher than that of another level, violated the second law of thermodynamics. This process does not proceed spontaneously, however. It is the interaction with polarised light that drives the system and increased disorder in the radiation field compensates for the increase in the population difference in the material system (Enk and Nienhuis 1992).

Rotational symmetry

Motivation

The number of energy levels that contribute to the dynamics of a quantum mechanical system is a direct measure of the number of degrees of freedom required for a full description of the system. The systems in which we are interested always include electric dipole moments – the degrees of freedom that couple to the radiation field. The second most important contribution is the magnetic dipole moment associated with the atomic angular momentum. Electric and magnetic dipole moments are those degrees of freedom that couple to external fields. Other degrees of freedom do not couple strongly to external fields but they may still modify the behaviour of the system and its optical properties.

Fundamentals

Motivation and principle

Motivation

The preceding chapter showed how light drives the internal dynamics of resonant atomic media and how the measurement of optical anisotropies allows us to monitor these dynamics. The experiments discussed in the preceding chapter, however, can provide only limited information about the system. Most physical systems have more degrees of freedom than we can observe by measurements on transmitted light. As another limitation, we have primarily considered atoms that evolve under their internal Hamiltonian, only weakly perturbed by the probe laser beam. The example of light-induced spin nutation showed that the dynamics of optically pumped atoms differ significantly from those of a free atom. Although it is possible to observe spin nutation for systems with more than two ground-state sublevels, such an experiment suffers from the damping that accompanies optical pumping. The damping drives the system rapidly to an equilibrium, too fast for detailed dynamical observations.

Light interacting with material substances is one of the prerequisites for life on our planet. More recently, it has become important for many technological applications, from CD players and optical communication to gravitational-wave astronomy. Physicists have therefore always tried to improve their understanding of the observed effects. The ultimate goal of such a development is always a microscopic description of the relevant processes. For a long time, this description was identical with a perturbation analysis of the material system in the external fields. More than a hundred years ago, such a microscopic theory was developed in terms of oscillating dipoles. After the development of quantum mechanics, these dipoles were replaced by quantum mechanical two-level systems, and this is still the most frequently used description.

However, the physical situation has changed qualitatively in the last decades. The development of intense, narrowband or pulsed lasers as tunable light sources has provided not only a new tool that allows much more detailed investigation, but also the observation of qualitatively new phenomena. These effects can no longer be analysed in the form of a perturbation expansion. One consequence is that the actual number of quantum mechanical states involved in the interaction becomes relevant. It is therefore not surprising that many newly discovered effects are associated with the details of the level structure of the medium used in the experiment. Two popular examples are the discovery of sub-Doppler laser cooling and the development of magnetooptical traps, which rely on the presence of angular momentum substates.

Phenomenological introduction

Model atoms

In the preceding section, we discussed the interaction between light and two-level atoms – probably one of the most popular physical models. The basis of this popularity is its intuitively simple interpretation combined with the posibility of explaining a wide range of physical phenomena. An interesting aspect of the two-level system is that, although its dynamics are formally equivalent to those of a classical angular momentum, it can explain many aspects of quantum mechanics. Once these aspects are understood, it is tempting to look further into the behaviour of real systems, trying to find patterns inconsistent with the predictions of the two-level model.

This section discusses aspects of the interaction between matter and radiation that are incompatible with the two-level model. We do not consider specific atomic systems or attempt a complete analysis of the dynamics that a three-level system can exhibit. Instead we select a number of phenomena that play an important role in the discussion of those physical systems that form the subject of the subsequent sections.

In view of the complexity of the various kinetic models discussed near the beginning of the previous chapter and of the vast array of cross sections and other atomic data required for their implementation, there is a great need for a more basic theory. Such a theory will have a restricted region of validity but can be used with confidence for the interpretation of plasma spectroscopic experiments performed under appropriate conditions. This basic theory is provided by equilibrium statistical mechanics or thermodynamics. It, together with the various conditions for its validity, is the subject of the present chapter. Before introducing this new topic, a second role for the more basic theory should be emphasized, namely its service in testing kinetic models under conditions for which thermodynamic equilibrium should hold.

The practical goal of any of the kinetic models and of the local thermodynamic equilibrium (LTE) relations introduced here is the calculation of populations of the various states of atoms or ions and of the free electron density for specified temperature(s), pressure or mass density, and assumed chemical composition. As already discussed in the preceding chapter, one deals with local, and usually also transient, equilibria, because strict thermodynamic equilibrium would either require an unbounded, spatially and temporally homogeneous plasma or a plasma enclosed in an ideal blackbody hohlraum (cavity).

In the preceding chapters, the basic radiative and collisional processes governing local radiative properties of a plasma were introduced; and their quantitative evaluation or acquisition was discussed to help the reader in the selection of data needed for analysis or prediction of a spectrum. We also learned about various kinetic or thermodynamic models designed for comprehensive descriptions of level populations.

There are two reasons for this seemingly all-encompassing body of theory and basic data to be insufficient, nevertheless, for both analysis and predictions. First, one generally cannot measure local values of the plasma emission, but must infer them from some distance and averaged over the various contributing volume elements. Second, and even more fundamentally, radiative processes also influence level populations so that the emission or absorption in one location depends on the radiation flux coming from the rest of the plasma.

Therefore a self-consistent treatment of radiation transport and level populations is necessary, requiring in general a nonlocal and nonlinear theory. Such theory has been developed over many years, mainly by astronomers and astrophysicists. Much progress has been made after the two basic treatises (Chandrasekhar 1950 and Sobolev 1963) were written, mostly by computational methods (see, e.g., Athay 1972, Kalkofen 1984, Crivellari, Hubeny and Hummer 1991), but also through more or less analytic models (see, e.g., Thomas and Athay 1961, Jefferies 1968, Ivanov 1973).

Depending on various plasma conditions, such as size, composition, densities and temperatures, the electron density and related quantities can be measured using a number of techniques. Of these, Langmuir probes (Tonks and Langmuir 1929, Hutchinson 1987) provided the first means to infer local values of the electron density, mostly at relatively low densities. Much more recently, Thomson scattering of laser light has become a method of choice for localized electron density measurements (Kunze 1968, Evans and Katzenstein 1969, DeSilva and Goldenbaum 1970, Sheffield 1975) over a range of about 1011 to 1021 cm−3. Then there are the usually inherently most accurate interferometric techniques, mostly laser-based as well (Jahoda and Sawyer 1971, Hauer and Baldis 1988), which span a similar range, recently extended to Ne ≈ 3 × 1021 cm−3 (DaSilva et al. 1995), but provide localized values of the electron density only indirectly. Using two or more wavelengths, one can, however, separate the free-electron contribution (1.37) to the refractive index from any boundstate contribution (2.103). In single-species, partially ionized plasmas, this boundstate contribution is a direct measure of the neutral atom density. Interferometric local values of the density can, in principle, be determined by methods analogous to those discussed for emission or absorption measurements in section 8.5.

In a fully ionized plasma, in which all bound electrons have been removed from their original atoms or molecules, there is besides the continuous spectrum no line emission or absorption, except possibly for features related to plasma resonances or waves (Bekefi 1966, Stix 1992, Swanson 1989). These often nonthermal features are usually at such low frequencies that they do not obscure or interfere with atomic radiation, the subject of principal interest here. Since atoms and incompletely stripped ions possess a continuous spectrum, besides the discrete spectrum providing the pairs of states involved in line radiation, continuous emission and absorption spectra underly and accompany the line spectra discussed in the preceding chapters. These processes are not only important as background to line emission, but also because continuum intensities can provide relatively direct measures of electron density and temperature (see sections 10.2 and 11.4, respectively).

With the usual convention for the energies of bound states as being negative relative to those of zero kinetic energy electrons at large distances from the nucleus of any isolated atom or ion, one might infer that all positive energy states belong to the continuous spectrum. In practice, this is an oversimplification, because there are states corresponding to the excitation of two or more bound electrons which are almost discrete.

Observable intensities of spectral lines and continua depend just as strongly on the appropriate level populations as on the transition probabilities and related quantities discussed in chapters 2 and 3, and on the radiative transfer processes to be discussed in chapter 8. The level population and transfer problems are not really separable. One must therefore strive for internal consistency, following the example of astrophysicists (see, e.g., Mihalas 1978). Nevertheless, it is reasonable to discuss first the various population and depopulation processes as if these interconnections were not very important. Often only processes affecting discrete states that are not very close to the corresponding continuum limits need to be evaluated explicitly. This important simplification arises because the rates of collisional processes controlling the relative populations of highly excited states tend to be very large, thus facilitating the establishment of so-called partial local thermodynamic equilibrium (PLTE, see sections 7.1 and 7.6). However, the lower excited level populations often need explicit evaluation, especially because these levels tend to give rise to the strongest lines. Note also that electron collisional rates usually, but not always, dominate because of their high velocities and of their energy-dependent cross sections that are similar to those for ions.